Tangent Vector

For a curve with Position Vector ${\bf r}(t)$, the unit tangent vector $\hat {\bf T}(t)$ is defined by

$$\hat {\bf T}(t) \equiv {{\bf r}'(t)\over \vert{\bf r}'(t)\vert} = {{d{\bf r}\over dt}\over \left\vert{d{\bf r}\over dt}\right\vert}$$ (1)

$$= {{d{\bf r}\over dt}\over {ds\over dt}}$$ (2)

$$= {d{\bf r}\over ds},$$ (3)

where $t$ is a parameterization variable and $s$ is the Arc Length. For a function given parametrically by $(f(t), g(t))$, the tangent vector relative to the point $(f(t), g(t))$ is therefore given by

$$x(t) = {f'\over\sqrt{f'^2+g'^2}}$$ (4)

$$y(t) = {g'\over\sqrt{f'^2+g'^2}}.$$ (5)

To actually place the vector tangent to the curve, it must be displaced by $(f(t), g(t))$. It is also true that

$${d\hat {\bf T}\over ds} = \kappa\hat {\bf N}$$ (6)

$${d\hat {\bf T}\over dt} = \kappa{ds\over dt}\hat {\bf N}$$ (7)

$${[}\dot{\bf T},\ddot{\bf T},\raise7.5pt\hbox{.}\mkern0mu\raise7.5pt\hbox{.}\mkern0mu\raise7.5pt\hbox{.}\mkern-15mu{\bf T}] = \kappa^5 {d\over ds}\left({\tau\over\kappa}\right),$$ (8)

where ${\bf N}$ is the Normal Vector, $\kappa$ is the Curvature, and $\tau$ is the Torsion.

See also Curvature, Normal Vector, Tangent, Tangent Bundle, Tangent Plane, Tangent Space, Torsion (Differential Geometry)

References

Gray, A. ``Tangent and Normal Lines to Plane Curves.'' §5.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 85-90, 1993.